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Painlevé Classification Problems Featuring Essential Singularities
Author(s) -
Cosgrove Christopher M.
Publication year - 1997
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00053
Subject(s) - gravitational singularity , mathematics , singularity , differential equation , regular singular point , logarithm , type (biology) , uniqueness , pure mathematics , generalization , mathematical analysis , ordinary differential equation , ecology , biology
In this article we construct and solve all Painlevé‐type differential equations of the second order and second degree that are built upon, in a natural well‐defined sense, the “sn‐log” equation of Painlevé, the general integral of which admits a movable essential singularity (elliptic function of a logarithm). This equation (which was studied by Painlevé in the years 1893–1902) is frequently cited in the modern literature to elucidate various aspects of Painlevé analysis and integrability of differential equations, especially the difficulty of detecting essential singularities by local singularity analysis of differential equations. Our definition of the Painlevé property permits movable essential singularities, provided there is no branching. While the essential singularity presents no serious technical problems, we do need to introduce new techniques for handling “exotic” Painlevé equations, which are Painlevé equations whose singular integrals admit movable branch points in the leading terms. We find that the corresponding full class of Painlevé‐type equations contains three, and only three, equations, which we denote SD‐326‐I, SD‐326‐II, and SD‐326‐III, each solvable in terms of elliptic functions. The first is Painlevé's own generalization of his sn‐log equation. The second and third are new, the third being a 15‐parameter exotic master equation. The appendices contain results (in general, without uniqueness proofs) of related Painlevé classification problems, including full generalizations of two other second‐degree equations discovered by Painlevé, additional examples of exotic Painlevé equations and Painlevé equations admitting movable essential singularities, and third‐order equations featuring sn‐log and other essential singularities.